Circles
Tangency conditions
Grade 11
Question:
<p>Given a line segment AB, A(0, 0) and B(a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the endpoints and the midpoint of the line segment AB. If \(0 < R < \frac{a}{4}\), then the number of possible circles S₄ that touch all 3 given circles is:</p>
<p>(a) 2</p>
<p>(b) 4</p>
<p>(c) 6</p>
<p>(d) 8</p>
Step-by-Step Solution
Key Concept: Consider both external and internal tangency conditions separately; geometric symmetry constrains the number of valid solutions.
<p><strong>Analysis:</strong> When \(0 < R < \frac{a}{4}\), the three circles are non-overlapping and positioned symmetrically on the x-axis. A fourth circle S₄ can be externally tangent to all three in multiple configurations: above the axis (external to all) and below the axis (external to all), plus internally tangent configurations. For this range of R, there are exactly 4 distinct circles S₄.</p>
Correct Answer: B